Throughout the book, several anti-windup constructions will be illustrated. They differ from each other in terms of applicability, namely, what control systems they can be applied to, stability and performance guarantees, and architecture. While most of the notation related to the concepts listed next will be discussed in Chapter 2, a general overview of all the algorithms is provided here, at the beginning of the book, as a quick reference for the different procedures and solutions available throughout.
Table 1 comparatively illustrates the applicability, architectures, and guarantees characterizing each algorithm (asterisks mean that some restrictions apply). The applicability is stated in terms of properties of the linear plant involved in the saturated control system, namely, exponentially stable (all the eigenvalues in the open left half plane), marginally stable (same as in the previous case, with possible single eigenvalues on the imaginary axis), marginally unstable (all the eigenvalues in the closed left half plane), and exponentially unstable (plants with at least one eigenvalue in the right half plane). This is not applicable for the last two algorithms because they address a special class of nonlinear plants. The architecture of the anti-windup solutions is characterized by their linear or nonlinear nature, the presence or not of dynamics in the anti-windup filter, and their interconnection properties: external or full authority. Finally, the guarantees on the compensated closed loop are distinguished as global or regional. For completeness, the page number where each algorithm appears is also listed in the second column of the table. It should be emphasized that all the direct linear anti-windup algorithms (namely, from Algorithm 1 to Algorithm 9) require linearity of the unconstrained controller to be applicable, whereas all the remaining model recovery anti-windup algorithms are applicable with any nonlinear controller and only linearity of the plant is required. Moreover, all MRAW algorithms correspond to plant-order, external augmentation.
A list of all the algorithm titles and their brief description is reported next, once again for a quick reference to the several solutions available in this book.
1. Static full-authority global DLAW (page 81): Simple architecture, very commonly used but not necessarily feasible for any exponentially stable plant. Global input-output gain is optimized.
2. Static external global DLAW (page 94): Useful when some internal states of the controller are inaccessible. Otherwise not more effective than Algorithm 1. Global input-output gain is optimized.
3. Static full-authority regional DLAW (page 99): Extends the applicability of Algorithm 1to a larger class of systems by requiring only regional properties. However, feasibility conditions still need to hold. Regional input-output gain is optimized.
Table 1 Applicability, architectures and guarantees of the algorithms illustrated in the book.
4. Dynamic plant-order full-authority global DLAW (page 114): Dynamic anti-windup, with state dimension equal to that of the plant, overcomes the applicability limitations of Algorithm 1. Feasible for any loop containing an exponentially stable plant. Global input-output gain is optimized; it is never worse, and often better, than the input-output gain provided by Algorithm 1.
5. Dynamic reduced-order full-authority global DLAW (page 126): Useful when static anti-windup is infeasible or performs poorly yet a low-order anti-windup augmentation is desired. Order reduction is carried out while maintaining a prescribed global input-output gain.
6. Dynamic plant-order external global DLAW (page 130): Useful when some internal states of the controller are inaccessible; otherwise not more effective than Algorithm 4. Feasible for any loop containing an exponentially stable plant. Global input-output gain is optimized; it is never worse, and often better, than the input-output gain provided by Algorithm 2.
7. Dynamic reduced-order external global DLAW (page 138): Useful when Algorithm 2 is not feasible or performs poorly yet a low-order external anti-windup augmentation is desired. Order reduction is carried out while maintaining a prescribed global input-output gain.
8. Dynamic plant-order full-authority regional DLAW (page 143): Extends the applicability of Algorithm 4 by only requiring regional properties. Applicable to any loop. Regional input-output gain is optimized.
9. Dynamic reduced-order full-authority regional DLAW (page 149): Useful when Algorithm 3 is not feasible or performs poorly yet a low-order anti-windup augmentation is desired. Order reduction is carried out while maintaining a prescribed global input-output gain.
10. Stability-based MRAW for exponentially stable plants (page 176): Special cases include IMC anti-windup and Lyapunov-based anti-windup, neither of which creates an algebraic loop. Global exponential stability is guaranteed but no performance measure is optimized.
11. Lyapunov-based MRAW for marginally stable plants (page 185): Does not create an algebraic loop. Global asymptotic stability is guaranteed but no performance measure is optimized.
12. Global LQ-based MRAW (page 189): A linear quadratic performance index related to URR is optimized subject to guaranteeing global exponential stability.
13. Global H2-based MRAW (page 194): An H2 performance index related to the unconstrained response recovery is optimized subject to guaranteeing global exponential stability.
14. Regional LQ-based MRAW (page 195): Extends the applicability of Algorithm 12 to any loop by requiring only regional exponential stability. Does not create an algebraic loop. A linear quadratic performance index related to small signal unconstrained response recovery is optimized.
15. Regional H2-based MRAW (page 197): Extends the applicability of Algorithm 13 to any loop by requiring only regional exponential stability. An H2 performance index related to small signal unconstrained response recovery is optimized.
16. Switched MRAW (page 203): Augmentation based on a family of nested ellipsoids and switching among a corresponding family of linear feedbacks. Regional unconstrained response recovery gain is optimized in each ellipsoid. Global exponential stability is guaranteed for any loop with an exponentially stable plant. Otherwise, regional exponential stability is guaranteed.
17. Scheduled MRAW (page 206): Augmentation based on a family of nested ellipsoids and continuous scheduling among a corresponding family of linear feedbacks. Regional unconstrained response recovery gain is optimized in each ellipsoid. Global exponential stability is guaranteed for any loop with an exponentially stable plant. Otherwise, regional exponential stability is guaranteed.
18. MPC-based MRAW with special terminal cost (page 215): Sampled-data augmentation based on model predictive control for constrained, discrete-time linear systems. An appropriate terminal cost function is used to guarantee global exponential stability for any loop with an exponentially stable plant. Regional exponential stability is guaranteed for any loop.
19. MPC-based MRAW with sufficiently long horizons (page 216): Sampled-data augmentation based on model predictive control for constrained, discrete-time linear systems. A sufficiently long optimization horizon is used to guarantee global exponential stability for any loop with an exponentially stable plant. Regional exponential stability is guaranteed for any loop.
20. Global MRAW using nested saturation (page 218): Augmentation using feedback consisting of multiple nested saturation functions. Does not create an algebraic loop. Global exponential stability is guaranteed for any loop with an exponentially stable plant. For a loop with a marginally unstable plant, global asymptotic stability is guaranteed. No performance measure is optimized.
21. Semiglobal MRAW using Riccati inequalities (page 221): Linear augmentation providing an arbitrarily large stability region for any loop containing a marginally unstable plant. Does not create an algebraic loop. Provides a quantified characterization of the regional unconstrained response recovery gain.
22. Global MRAW using scheduled Riccati inequalities (page 221): Augmentation based on a family of Riccati inequalities and continuous scheduling among a corresponding family of linear feedbacks. Global exponential stability is guaranteed for any loop with an exponentially stable plant. Global asymptotic stability is guaranteed for any loop with a marginally unstable plant.
23. MRAW with guaranteed region of attraction (page 223): Augmentation that, in order to achieve a large operating region for a loop containing an exponentially unstable plant, uses measurements of the plant's exponentially unstable modes.
24. MRAW for Euler-Lagrange systems with linear injection (page 248): Augmentation for a loop containing a plant from a class of nonlinear systems. Global asymptotic stability is guaranteed for any loop containing a plant whose parameters satisfy a certain constraint.
25. MRAW for Euler-Lagrange systems with nonlinear injection (page 249): Extends and improves upon the performance in Algorithm 24 by replacing certain linear terms with nonlinear terms.